Relation between metric spaces and Finsler spaces

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Abstract

In a connected Finsler space Fn = (M, F) every ordered pair of points p, q ∈ M determines a distance ρ{variant}F (p, q) as the infimum of the arc length of curves joining p to q. (M, ρ{variant}F) is a metric space if Fn is absolutely homogeneous, and it is quasi-metric space (i.e. the symmetry: ρ{variant}F (p, q) = ρ{variant}F (q, p) fails) if Fn is positively homogeneous only. It is known the Busemann-Mayer relation limt → t0+ frac(d, d t) ρ{variant}F (p0, p (t)) = F (p0, over(p, ̇)0), for any differentiable curve p (t) in an Fn. This establishes a 1 : 1 relation between Finsler spaces Fn = (M, F) and (quasi-) metric spaces (M, ρ{variant}F). We show that a distance function ρ{variant} (p, q) (with the differentiability property of ρ{variant}F) needs not to be a ρ{variant}F. This means that the family {(M, ρ{variant})} is wider than {(M, ρ{variant}F)}. We give a necessary and sufficient condition in two versions for a ρ{variant} to be a ρ{variant}F, i.e. for a (quasi-) metric space (M, ρ{variant}) to be equivalent (with respect to the distance) to a Finsler space (M, F). © 2008 Elsevier B.V. All rights reserved.

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Tamássy, L. (2008). Relation between metric spaces and Finsler spaces. Differential Geometry and Its Application, 26(5), 483–494. https://doi.org/10.1016/j.difgeo.2008.04.007

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